Compressing information as much as possible and in the shortest time is a fundamental challenge for communications, as well as for data storage and processing. As information grows, it becomes essential to use efficient methods that optimize space and speed without losing quality.

Data compression not only allows faster and more cost-effective handling of information but also facilitates its secure transmission between systems and devices. In this section, we will cover advanced techniques, including recursive compression, and explore how to apply them to improve efficiency in modern environments.

The concept of reducing information dates back to the earliest coding and data transmission systems:

• 1940s–50s: Huffman and Shannon-Fano codes establish the mathematical foundation for compression.
• 1970s–80s: Standards such as ZIP and GIF are developed, making file compression accessible to everyday users.
• Digital era (1990s onward): MPEG, JPEG, and MP3 revolutionize audio, video, and image compression, adapting to modern multimedia.

Today, compression continues to evolve to handle massive data volumes and real-time applications.

There are mainly two approaches:

1. Lossless compression: allows exact reconstruction of the original data. Examples: ZIP, PNG, FLAC.
2. Lossy compression: irreversibly reduces information to achieve greater compression, useful in multimedia. Examples: JPEG, MP3, H.264.

Both types use sophisticated algorithms that search for patterns, redundancies, and repeated structures within the data.

Compression can be understood as an efficient coding problem:

• Each dataset has an entropy, measuring the minimum number of bits required to represent it.
• Compression algorithms aim to eliminate redundancy while preserving data integrity or desired quality.

A simple example of lossless compression is Run-Length Encoding (RLE), which replaces repeated sequences with a value and its count:

  def rle_encode(data):
      """Encode data using Run-Length Encoding"""
      encoding = []
      prev_char = data[0]
      count = 1
      for char in data[1:]:
          if char == prev_char:
              count += 1
          else:
              encoding.append((prev_char, count))
              prev_char = char
              count = 1
      encoding.append((prev_char, count))
      return encoding

  function rleEncode(data) {
      // Encode data using Run-Length Encoding
      const encoding = [];
      let prevChar = data[0];
      let count = 1;
      
      for (let i = 1; i < data.length; i++) {
          if (data[i] === prevChar) {
              count++;
          } else {
              encoding.push([prevChar, count]);
              prevChar = data[i];
              count = 1;
          }
      }
      encoding.push([prevChar, count]);
      return encoding;
  }

  std::vector<std::pair<char, int>> rleEncode(const std::string& data) {
      // Encode data using Run-Length Encoding
      std::vector<std::pair<char, int>> encoding;
      char prevChar = data[0];
      int count = 1;
      
      for (size_t i = 1; i < data.length(); i++) {
          if (data[i] == prevChar) {
              count++;
          } else {
              encoding.push_back({prevChar, count});
              prevChar = data[i];
              count = 1;
          }
      }
      encoding.push_back({prevChar, count});
      return encoding;
  }

This example shows how small mathematical optimizations can produce significant space savings.

Compression is essential in multiple areas:

• Cloud storage: reduces costs and speeds up access to large volumes of data.
• Video and audio streaming: allows real-time content delivery with limited bandwidth.
• Networks and communications: enables efficient transfer of messages and files between connected devices.
• Artificial intelligence and big data: improves the handling of massive datasets and trained models.

Advanced compression, including recursive compression, allows combining methods to maximize efficiency benefits.

To better understand compression, practical exercises can be performed:

1. Test RLE with text strings: analyze how size changes when compressing repeated sequences.
2. Compare lossless vs lossy compression: compress images using PNG and JPEG and observe differences in quality and size.
3. Mini-project on recursive compression: apply an algorithm like ZIP on a set of already compressed files and analyze results.

These exercises demonstrate how compression combines logic, mathematics, and technical creativity.

Compression is history, mathematics, and applied technology. From the first codes to modern methods, its goal has always been to maximize efficiency in information handling.

In a world where data grows exponentially, compression allows space savings, faster processing, and efficient transmissions, becoming an essential tool for all modern technology.